Generalized Ramsey numbers via conflict-free hypergraph matchings

Given graphs $G, H$ and an integer q ≥ 2, the generalized Ramsey number, denoted $r(G,H,q)$, is the minimum number of colours needed to edge-colour G such that every copy of H receives at least q colours. In this paper, we prove that for a fixed integer k ≥ 3, we have r(Kₙ,Cₖ,3) = n/(k-2)+o(n). This generalises work of Joos and Muybayi, who proved r(Kₙ,C₄,3) = n/2+o(n). We also provide an upper bound on r(Kn,n, Cₖ, 3), which generalises a result of Joos and Mubayi that r(Kn,n,C₄,3) = 2n/3+o(n). Both of our results are in fact specific cases of more general theorems concerning families of cycles.